Properties

Label 9408.2.a.ds
Level $9408$
Weight $2$
Character orbit 9408.a
Self dual yes
Analytic conductor $75.123$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9408 = 2^{6} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9408.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(75.1232582216\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( 2 + \beta ) q^{5} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 2 + \beta ) q^{5} + q^{9} + ( 2 + 2 \beta ) q^{11} + \beta q^{13} + ( -2 - \beta ) q^{15} + ( 2 + 3 \beta ) q^{17} + ( -4 + 2 \beta ) q^{19} + ( -2 + 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{25} - q^{27} + 6 \beta q^{29} + ( 8 + 2 \beta ) q^{31} + ( -2 - 2 \beta ) q^{33} + ( 4 + 4 \beta ) q^{37} -\beta q^{39} + ( 2 + \beta ) q^{41} -8 q^{43} + ( 2 + \beta ) q^{45} + ( 4 - 2 \beta ) q^{47} + ( -2 - 3 \beta ) q^{51} + ( 2 - 8 \beta ) q^{53} + ( 8 + 6 \beta ) q^{55} + ( 4 - 2 \beta ) q^{57} + ( -8 - 2 \beta ) q^{59} + ( 4 - 7 \beta ) q^{61} + ( 2 + 2 \beta ) q^{65} -8 q^{67} + ( 2 - 2 \beta ) q^{69} + ( -2 - 2 \beta ) q^{71} + ( 4 - 5 \beta ) q^{73} + ( -1 - 4 \beta ) q^{75} + ( 8 + 4 \beta ) q^{79} + q^{81} + ( -4 + 8 \beta ) q^{83} + ( 10 + 8 \beta ) q^{85} -6 \beta q^{87} + ( -2 - 9 \beta ) q^{89} + ( -8 - 2 \beta ) q^{93} -4 q^{95} + ( 12 + 3 \beta ) q^{97} + ( 2 + 2 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{5} + 2q^{9} + 4q^{11} - 4q^{15} + 4q^{17} - 8q^{19} - 4q^{23} + 2q^{25} - 2q^{27} + 16q^{31} - 4q^{33} + 8q^{37} + 4q^{41} - 16q^{43} + 4q^{45} + 8q^{47} - 4q^{51} + 4q^{53} + 16q^{55} + 8q^{57} - 16q^{59} + 8q^{61} + 4q^{65} - 16q^{67} + 4q^{69} - 4q^{71} + 8q^{73} - 2q^{75} + 16q^{79} + 2q^{81} - 8q^{83} + 20q^{85} - 4q^{89} - 16q^{93} - 8q^{95} + 24q^{97} + 4q^{99} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.41421
1.41421
0 −1.00000 0 0.585786 0 0 0 1.00000 0
1.2 0 −1.00000 0 3.41421 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9408.2.a.ds 2
4.b odd 2 1 9408.2.a.ee 2
7.b odd 2 1 9408.2.a.du 2
8.b even 2 1 2352.2.a.bd 2
8.d odd 2 1 1176.2.a.j 2
24.f even 2 1 3528.2.a.bl 2
24.h odd 2 1 7056.2.a.cx 2
28.d even 2 1 9408.2.a.dg 2
56.e even 2 1 1176.2.a.o yes 2
56.h odd 2 1 2352.2.a.bb 2
56.j odd 6 2 2352.2.q.be 4
56.k odd 6 2 1176.2.q.o 4
56.m even 6 2 1176.2.q.k 4
56.p even 6 2 2352.2.q.bc 4
168.e odd 2 1 3528.2.a.bb 2
168.i even 2 1 7056.2.a.cg 2
168.v even 6 2 3528.2.s.bd 4
168.be odd 6 2 3528.2.s.bm 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 8.d odd 2 1
1176.2.a.o yes 2 56.e even 2 1
1176.2.q.k 4 56.m even 6 2
1176.2.q.o 4 56.k odd 6 2
2352.2.a.bb 2 56.h odd 2 1
2352.2.a.bd 2 8.b even 2 1
2352.2.q.bc 4 56.p even 6 2
2352.2.q.be 4 56.j odd 6 2
3528.2.a.bb 2 168.e odd 2 1
3528.2.a.bl 2 24.f even 2 1
3528.2.s.bd 4 168.v even 6 2
3528.2.s.bm 4 168.be odd 6 2
7056.2.a.cg 2 168.i even 2 1
7056.2.a.cx 2 24.h odd 2 1
9408.2.a.dg 2 28.d even 2 1
9408.2.a.ds 2 1.a even 1 1 trivial
9408.2.a.du 2 7.b odd 2 1
9408.2.a.ee 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9408))\):

\( T_{5}^{2} - 4 T_{5} + 2 \)
\( T_{11}^{2} - 4 T_{11} - 4 \)
\( T_{13}^{2} - 2 \)
\( T_{17}^{2} - 4 T_{17} - 14 \)
\( T_{19}^{2} + 8 T_{19} + 8 \)
\( T_{31}^{2} - 16 T_{31} + 56 \)